Cont and Fournié ([1]}, [2]}, [3]}) as well as Levental et al. ([4]}) used different formalizations of Dupire's idea to derive versions of a functional Itō-formula for real-valued semi-martingales using different derivatives. In addition, Cont used the functional Itō-formula to derive a martingale representation formula for Brownian martingales and extended the functional derivative to a weak derivative for square-integrable functionals (see [2]}, [3]}).
